The bomber approached the crowd of dozens on foot in provincial capital Jalalabad as they were demonstrating in support of a local police commander who had been sacked and calling for his reinstatement, provincial spokesman Attaullah Khogyani said.
"Unfortunately, eight people were killed, all of them civilians," he said.
A further 15 people including children were also wounded in the explosion, he added.
The casualty toll was also confirmed by provincial health director Najib Kamawal, who said some of those wounded were in a serious condition.
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But the Taliban are active in Nangarhar province, as are the Islamic State group.
Afghan and US officials have stepped up their attacks against IS and recently said it is steadily losing territory, with fighters largely confined to two or three districts in Nangarhar from around nine in January.
But the group has intensified attacks across the country, particularly in the east and in the capital Kabul, recruiting followers and in some places challenging the Taliban on its own turf.